Question

Use vertical multiplication to find the product of:


`x ^(3) + 2x + 3 * x ^(3) - x + 1`

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Answer 1

`x^6+x^4+4x^3-2x^2-x+3`

Explanation:

`x^3+2x+3`

`*(x^3-x+1)`

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First step multiply your terms in your first expression just to the 1 in the second expression like so:

`x^3+2x+3`

`*(x^3-x+1)`

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`x^3+2x+3` Anything times 1 is that anything.

That is,
`(x^3+2x+3) \cdot 1=x^3+2x+3`.

Now we are going to take the top expression and multiply it to the -x in the second expression.
`-x(x^3+2x+3)=-x^4-2x^2-3x`. We are going to put this product right under our previous product.

`x^3+2x+3`

`*(x^3-x+1)`

---------------------------------

`x^3+2x+3`

`-x^4-2x^2-3x`

We still have one more multiplication but before we do that I'm going to put some 0 place holders in and get my like terms lined up for the later addition:

`x^3+2x+3`

`*(x^3-x+1)`

---------------------------------

`0x^4+x^3+0x^2+2x+3`

`-x^4+0x^3-2x^2-3x+0`

Now for the last multiplication, we are going to take the top expression and multiply it to x^3 giving us
`x^3(x^3+2x+3)=x^6+2x^4+3x^3`. (I'm going to put this product underneath our other 2 products):

`x^3+2x+3`

`*(x^3-x+1)`

---------------------------------

`0x^4+x^3+0x^2+2x+3`

`-x^4+0x^3-2x^2-3x+0`

`x^6+2x^4+3x^3`

I'm going to again insert some zero placeholders to help me line up my like terms for the addition.

`x^3+2x+3`

`*(x^3-x+1)`

---------------------------------

`0x^6+0x^4+x^3+0x^2+2x+3`

`0x^6-x^4+0x^3-2x^2-3x+0`

`x^6+2x^4+3x^3+0x^2+0x+0`

----------------------------------------------------Adding the three products!

`x^6+x^4+4x^3-2x^2-x+3`